Generalized twisted sectors of orbifolds

For a finitely generated discrete group $\Gamma$, the $\Gamma$-sectors of an orbifold $Q$ are a disjoint union of orbifolds corresponding to homomorphisms from $\Gamma$ into a groupoid presenting $Q$. Here, we show that the inertia orbifold and $k$-multi-sectors are special cases of the $\Gamma$-sectors, and that the $\Gamma$-sectors are orbifold covers of Leida's fixed-point sectors. In the case of a global quotient, we show that the $\Gamma$-sectors correspond to orbifolds considered by other authors for global quotient orbifolds as well as their direct generalization to the case of an orbifold given by a quotient by a Lie group. Furthermore, we develop a model for the $\Gamma$-sectors corresponding to a generalized loop space.